Ensemble Quantum Research Articles

Parallel quantum computing in a single ensemble quantum computer

We propose a parallel quantum computing mode for ensemble quantum computer. In this mode, some qubits are in pure states while other qubits are in mixed states. It enables a single ensemble quantum computer to perform ``single-instruction-multidata'' type of parallel computation. Parallel quantum computing can provide additional speedup in Grover's algorithm and Shor's algorithm. In addition, it also makes a fuller use of qubit resources in an ensemble quantum computer. As a result, some qubits discarded in the preparation of an effective pure state in the Schulman-Varizani and the Cleve-DiVincenzo algorithms can be reutilized.

Designing robust gate implementations for quantum-information processing

Quantum-information processing systems are often operated through time-dependent controls; choosing these controls in a way that makes the resulting operation insensitive to variations in unknown or uncontrollable system parameters is an important prerequisite for obtaining high-fidelity gate operations. In this article we present a numerical method for constructing such robust control sequences for a quite general class of quantum-information processing systems. As an application of the method we have designed a robust implementation of a phase-shift operation central to rare-earth-metal quantum computing, an ensemble quantum computing system proposed by Ohlsson et al. [Opt. Commun. 201, 71 (2002)]. In this case the method has been used to obtain a high degree of insensitivity with respect to differences between ensemble members, but it is equally well suited for quantum computing with a single physical system.

Experimental realization of a fetching algorithm in a 7-qubit NMR spin Liouville space computer

Using a 7-qubit spin Liouville space computer, we have demonstrated successfully a fetching algorithm that finds marked items using only a single query. The essential idea is the operation of quantum computers in classical parallel. We gave the detailed pulse sequence for coherent control of the 7 qubits. A generalization of the pulse sequences for arbitrary qubit number is also provided.

Estimating Bohm’s quantum force using Bayesian statistics

In this paper we develop an approximate methodológy for estimating the multidimensional quantum density associated with a statistical bundle of de Broglie–Bohm trajectories. The quantum density is constructed as a discrete sum of nonequivalent Gaussian components. We incorporate the ideas of Bayesian statistical analysis and an expectation-maximization procedure to compute an approximate quantum force that drives the statistical ensemble quantum trajectories.

Entanglement constrained by superselection rules.

Bipartite entanglement may be reduced if there are restrictions on allowed local operations. We introduce the concept of a generalized superselection rule to describe such restrictions, and quantify the entanglement constrained by it. We show that ensemble quantum information processing, where elements in the ensemble are not individually addressable, is subject to the superselection rule associated with the symmetric group (the group of permutations of elements). We prove that even for an ensemble comprising many pairs of qubits, each pair described by a pure Bell state, the entanglement per element constrained by this superselection rule goes to zero for a large number of elements.

Quantum Mechanical Nature in Liquid NMR Quantum Computing**The project supported by National Fundamental Research Program under Grant No. 001CB309308, and National Natural Science Foundation of China under Grant No. 60073009, the Hang Tian Science Fund, and the Excellent Young University Teachers' Fund of the Education Ministry of China

The quantum nature of bulk ensemble NMR quantum computing — the center of recent heated debate, is addressed. Concepts of the mixed state and entanglement are examined, and the data in a two-qubit liquid NMR quantum computation are analyzed. The main points in this paper are: i) Density matrix describes the “state” of an average particle in an ensemble. It does not describe the state of an individual particle in an ensemble; ii) Entanglement is a property of the wave function of a microscopic particle (such as a molecule in a liquid NMR sample), and separability of the density matrix cannot be used to measure the entanglement of mixed ensemble; iii) The state evolution in bulk-ensemble NMR quantum computation is quantum-mechanical; iv) The coefficient before the effective pure state density matrix, є, is a measure of the simultaneity of the molecules in an ensemble. It reflects the intensity of the NMR signal and has no significance in quantifying the entanglement in the bulk ensemble NMR system. The decomposition of the density matrix into product states is only an indication that the ensemble can be prepared by an ensemble with the particles unentangled. We conclude that effective-pure-state NMR quantum computation is genuine, not just classical simulations.

Fluorescence quantum yield of CdSe/ZnS nanocrystals investigated by correlated atomic-force and single-particle fluorescence microscopy

Correlated atomic-force and fluorescence microscopy are used to study single-particle versus ensemble fluorescence quantum yields (QY) of semiconductor nanocrystals by measuring a simultaneous map of the topography and the single-particle fluorescence. CdSe/ZnS nanocrystal quantum dots and quantum rods with high QY were investigated. A significant portion of dark particles is detected. Comparison with the ensemble solution QY shows that samples with higher QY have a larger fraction of bright particles accompanied by an increased single-particle QY. Saturated emission from single nanocrystals could not be detected because of particle darkening under high-power excitation.

Laser phase modulation approaches towards ensemble quantum computing.

Selective control of decoherence is demonstrated for a multilevel system by generalizing the instantaneous phase of any chirped pulse as individual terms of a Taylor series expansion. In the case of a simple two-level system, all odd terms in the series lead to population inversion, while the even terms lead to self-induced transparency. These results also hold for multiphoton transitions that do not have any lower-order photon resonance or any intermediate virtual state dynamics within the laser pulse width. Such results form the basis of a robustly implementable CNOT gate.

A simulator for ensemble quantum computing

A quantum computer simulator based on the principles of ensemble quantum computing using nuclear magnetic resonance (NMR) is presented. The first version of ensemble quantum computer simulator (enQC-lator) is based on density matrix description of the NMR and models the implementation of quantum computation in a NMR spectrometer. The simulator is designed to perform general purpose quantum computing and will be useful to evaluate quantum-computing codes prior to their actual implementation and to design new ones. The performance of the simulator is demonstrated in several logical operations and algorithms.

Estimating the efficiency of ensemble quantum computing

A straightforward method to estimate the efficiency of ensemble quantum computing using nuclear magnetic resonance spectroscopy is presented by defining a quantum computing efficiency factor (QC ε ). This method allows critical evaluation of the performance of quantum gates and algorithms and provides a way to design efficient quantum computer.

Implementation of a quantum algorithm to solve the Bernstein-Vazirani parity problem without entanglement on an ensemble quantum computer

Bernstein and Varizani have given the first quantum algorithm to solve the parity problem in which a strong violation of the classical information theoretic bound comes about. In this paper we refine this algorithm with fewer resources and implement a two-qubit algorithm in a single query on an ensemble quantum computer.

Quantum Computing Using a Neutral Atom Optical Lattice: An Appraisal

We present a review of the neutral atom optical lattice quantum computer [Brennen et al., Phys. Rev. Lett. 82, 1060 (1999)] , which uses a laser mediated dipole-dipole coupling between the nearby atoms in an optical lattice to effect a quantum logic gate. Effectively the model creates a periodic array of two qubit (diatomic) systems, and should be considered as a kind of ensemble quantum computer operating at very low temperature.

Quantum simulation with natural decoherence

A quantum system may be efficiently simulated by a quantum information processor as suggested by Feynman and developed by Lloyd, Wiesner, and Zalka. Within the limits of the experimental implementation, simulation permits the design and control of the kinematic and dynamic parameters of a quantum system. Extension to the inclusion of the effects of decoherence, if approached from a full quantum-mechanical treatment of the system and the environment, or from a semiclassical fluctuating field treatment (Langevin), requires the difficult access to dynamics on the time scale of the environment correlation time. Alternatively, a quantum-statistical approach may be taken which exploits the natural decoherence of the experimental system, and requires a more modest control of the dynamics. This is illustrated for quantum simulations of a four-level quantum system by a two-spin NMR ensemble quantum information processor.

Use of composite rotations to correct systematic errors in NMR quantum computation

We implement an ensemble quantum counting algorithm on three NMR spectrometers with 1H resonance frequencies of 500, 600 and 750 MHz. At higher frequencies, the results deviate markedly from naive theoretical predictions. These systematic errors can be attributed almost entirely to off-resonance effects, which can be substantially corrected for using fully compensating composite rotation pulse sequences originally developed by Tycko. We also derive an analytic expression for generating such sequences with arbitrary rotation angles.

The relationship between ensemble quantum computing logical gates and NMR pulse sequence engineering exemplified by the SWAP operation

This paper describes how to derive nuclear magnetic resonance (NMR) pulse sequences suitable as ensemble quantum computing logical gates by known systematic tools for design of pulse sequences in NMR spectroscopy. This is in contrast to the current trend of developing NMR pulse sequences for ensemble quantum computing independent of established NMR spectroscopic technology, although many of the pertinent spin manipulations are already known in NMR spectroscopy. The SWAP operation is discussed in detail and one version is only (2J)−1long, which compares favorably with 3(2J)−1 in a recent implementation (Linden N, Barjat H, Kupce E, Freeman R. Chem Phys Lett 1999; 307:198–204). Alternatively, the number of π pulses can be reduced in that sequence. © 2000 John Wiley & Sons, Inc. Concepts Magn Reson 12: 389–395, 2000

The relationship between ensemble quantum computing logical gates and NMR pulse sequence engineering exemplified by the SWAP operation

The relationship between ensemble quantum computing logical gates and NMR pulse sequence engineering exemplified by the SWAP operation

Construction and Implementation of NMR Quantum Logic Gates for Two Spin Systems

The implementation of small prototype quantum computers has been studied through ensemble quantum computing via NMR measurements. In such laboratory studies it is convenient to have access to a wide array of logic gates. Here a systematic approach to reduce the logic gate to an NMR pulse sequence is introduced. This approach views the truth table for a quantum logic operation as a permutation matrix that corresponds to a propagator for an NMR transition. This propagator is then used as the starting point for the derivation of a pulse sequence. Pulse sequences for all the permutations of a four level system are reported along with implementations of representative examples on a two spin-12 system, 13C-labeled chloroform.

Approximate Quantum Counting on an NMR Ensemble Quantum Computer

We demonstrate the implementation of a quantum algorithm for estimating the number of matching items in a search operation using a two qubit nuclear magnetic resonance (NMR) quantum computer.

Sub-Riemannian geometry and clinical magnetic resonance tomography

Mensch, streckh deine Vernunfft hieher, diese dinge zu begreiffen! Johann Keppler (1571–1630) There is nothing that nuclear spins will not do for you, as long as you treat them as human beings. Erwin Louis Hahn (1949) Ein maschinelles ‘agencement’ ist den Schichten zugewandt, reinen Intensitaten, die sie zirkulieren last um die Selektion der ‘Konsistenzebene’ zu sichern, und der sich die Subjekte zuordnen, welchen sie einen Namen nur als Spur einer Intensitat last. Gilles Deleuze und Felix Guattari (1988) The sense we make of the world is governed by our conceptual means. More specifically, sense is made of nature by projecting a conceptual structure onto observed events. This is accomplished by means of a semantic filter, which transforms raw data into semantically significant events. George L. Farre (1997) Phase coherent wavelets form a unified basis of the multichannel reconstruction analysis–synthesis filter bank, implemented by high-resolution synthetic aperture radar (SAR) imaging and clinical MRI. The construction of unitary bank filters is performed by the Kepplerian stratification technique. It allows for the cross-sectional quadrature filtering of phase histories in contiguous local frequency encoding subband multichannels relative to the rotating coordinate frame of quadrature reference. The Kepplerian quadrature detection strategy, which is based on the unitary filter bank construction takes place in symplectic affine planes of incidence immersed into the three-dimensional compact super-encoding projective space. The planes are implemented in the quadrature format of a phase-splitting network of Fourier analysis of the stratified Heisenberg nilpotent Lie group G. The action of the basic Lie group G of quantum physics under its sub-Riemannian geometry admits a matrix coordinatization by transvections. In terms of projective geometry, the longitudinal dilations act as a spectral transform on the transvections of the G-action. The tomographic slices, frequency selected by the MRI scanner system, are identified with the projectively immersed, symplectic affine leaves ν (ν=0) of the canonical coadjoint orbit foliation of G, on which the continuous affine wavelet transform lifts to a transversal spectral transform. The paper points out that the key point of ensemble quantum computation by Fourier transform MRI is the synergy of the various wavelets, and the coexistence of their geometric substrates recorded by the bi-infinite stratigraphic time line. The paper leads from Keppler to Heisenberg and emphasizes the tight control which Lie groups exercise over the non-invasive MRI modality via geometric quantization. The fascinating aspects of electronic engineering concerned with the calculation of the Weyl symbol are also considered. Copyright © 1999 John Wiley & Sons, Ltd.

One- and two-dimensional ensemble quantum computing in spin Liouville space

New concepts are described for nuclear magnetic resonance (NMR) implementations of spin ensemble quantum computing in one and two dimensions. Similarities and differences between ensemble and pure state quantum computing are discussed by using a Liouville space formalism based on polarization and single transition operators. The introduction of an observer spin, that is coupled to the spins carrying the quantum bits, allows a mapping of the states of a quantum computer on a set of transitions between energy levels. This is spectroscopically favorable compared to a mapping on the energy levels themselves. Two complementary parallelization schemes for quantum computing are presented: one exploits the parallel processing feature inherent in multidimensional NMR, while the other employs mixed superposition states represented by operators in Liouville space. The spin swap operation, introduced in this paper, allows a convenient extension of quantum computing to spin systems where not all spin–spin couplings are resolved. The concepts are illustrated by implementations of logic operations and identities consisting of a sequence of basic logic gate operations.